infinite sequence

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An infinite sequence is a list or string of discrete objects, usually numbers, that can be paired off one-to-one with the set of positive integer s {1, 2, 3, ...}. Examples of infinite sequences are N = (0, 1, 2, 3, ...) and S = (1, 1/2, 1/4, 1/8, ..., 1/2 n , ...). The fact that a sequence is infinite is indicated by three dots following the last listed member.

An infinite series is the sum of the values in an infinite sequence of numbers. In the above examples, the sum of the numbers in N is the series n = 0 + 1 + 2 + 3 + ..., which is is undefined. But the sum of the numbers in S is the series s = 1 + 1/2 + 1/4 + 1/8 + ... + 1/2 n + ..., which is defined and equal to 2. When the sum of an infinite series is finite and definable, then that series and its corresponding seqeuence converge. Otherwise, the series and its corresponding sequence diverge.

Infinite sequences and series are important in physics and engineering. One of the most well-known is the Fourier series , which can mathematically define certain signal waveform s. Non-mathematicians often use the term series when they mean sequence. Technically, a series is always the sum of the numbers in a specific sequence. An infinite series is the sum, if defined, of the numbers in a specific infinite sequence.

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